The short version: the process we call "burning" is a chemical reaction in which complex, high-energy molecules (oil, wood, etc.) combine with oxygen to form lower-energy molcules such as carbon dioxide and water. The "extra" energy no longer stored in the chemical bonds of the fuel is released into the environment as heat. What we perceive as "flame" is the region in which this reaction goes on.

A little more info: the reason why flames spread is that a great number of molecules are not in their lowest-energy state (which is, thermodynamically, the most stable - where they are "happiest"). If all molecules immediately went into their lowest energy state, we (and most other things on Earth) would all spontaneously burn up and life wouldn't exist. Fortunately, many reactions will only proceed if the reacting molecules have a minimum kinetic energy (i.e. speed with which they're moving around) in order to proceed. In practice, this means that most combustion reactions won't go until the fuel is heated up. However, once combustion starts, it releases large quantities of heat into the environment, and this released heat gives neighboring molecules the kinetic energy necessary for them to burn, too. The result is that once a fire starts, it can spread easily . . .

A last fun tidbit - some of the things we consider inflammable don't burn because they are already in a low-energy state (for example, water). Other things are capable of burning, but don't burn easily because they have a high energy barrier; certain metals will burn spectacularly if you heat them up enough. A classic example of this is thermite, which is just aluminum and rust (iron oxide) - a blowtorch often won't heat these up enough to start the reaction, but once combustion begins, it's extremely intense.